Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects the distance it travels at each stage.

Speeding Up, Slowing Down

Stage: 3 Challenge Level:

Why do this problem?

This
problem provides a visual context in which to consider how
speed / time graphs represent movement over time. It allows
opportunity for learners to discuss and refine their ideas. Asking
learners to predict, to justify their predictions and to consider
modifying their views can help address misconceptions and improve
understanding. This problem could be used together with How
far does it move?

Possible approach

With the interactivity in its initial settings (i.e. dot in
the centre) ask group to predict what the path of the red dot will
be and what the speed-time graph will look like. Students could
sketch the path and graph in advance, before seeing the polygon
roll. Their suggestions could be compared and discussed before
making a final joint judgement on the shapes of the path and the
graph.

Run the interactivity. Discuss how the graph related to what
students expected. Confirm understanding by asking what would
happen if you changed the polygon to a triangle, a square, ... or a
circle.

At this point you might choose to do this card sorting activity
which looks at all the possible positions for the starting point
for the dot on a triangle. Can the students match the triangle with
its corresponding graph and locus?

When the group feel confident, move them on to more
challenging situations by moving the red point to a vertex of a
pentagon. Ask similar questions about the path of the red dot and
the speed-time graph.

Allow plenty of time for discussing/comparing different ideas
before running the interactivity. The pause button is useful to
focus on the different stages of the journey and to ask for
conjectures about what will follow.

Ask pairs or groups to work on new questions, agreeing and
drawing the graph and path together before using the interactivity
to confirm their ideas.

Suitable questions are:

What happens if the dot is moved to a different vertex?

What happens if the dot is placed on a vertex of a different
polygon?

What happens if the dot is in the middle of a side of a
polygon?

Key questions

Why does the graph jump up or down (i.e. why are there
discontinuities)?

Why does the dot speed up and slow down at different stages of
the "journeys"?

If we change - (the polygon/position of dot) - what will be the
same about the graph and what will be different?

Possible extension

Pairs could draw out a speed-time graph they have generated
and post it as a challenge for others to establish which initial
settings were used(in the least number of guesses).

Possible support

There are a number of variables affecting the speed /time
graphs. Fix all but one of the variables and spend time making
sense of the impact of that variable. For example, keeping the dot
fixed at a vertex and examining what happens for the same polygon
but for different sizes, then for different vertices before moving
onto different polygons.

The card sorting activity mentioned above keeps the polygon
fixed as a triangle. Other card sorting activities could be
produced.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.